91Ó°ÊÓ

Find \(u \cdot v,\) where \(\theta\) is the angle between u and v. $$\|\mathbf{u}\|=100,\|\mathbf{v}\|=250, \theta=\frac{\pi}{6}$$

Multiplying or Dividing Complex Numbers Perform the operation and leave the result in trigonometric form. $$\frac{3\left(\cos 50^{\circ}+i \sin 50^{\circ}\right)}{9\left(\cos 20^{\circ}+i \sin 20^{\circ}\right)}$$

Find \(u \cdot v,\) where \(\theta\) is the angle between u and v. $$\|\mathbf{u}\|=9,\|\mathbf{v}\|=36, \quad \theta=\frac{3 \pi}{4}$$

Find the vector \(\mathbf{v}\) with the given magnitude and the same direction as \(\mathbf{u}\). $$\begin{array}{cc}\text{Magnitude} && \text{Direction} \\ \|\mathbf{v}\|=9 && \mathbf{u}=\langle 2,5\rangle \end{array}$$

Multiplying or Dividing Complex Numbers Perform the operation and leave the result in trigonometric form. $$\frac{\cos 120^{\circ}+i \sin 120^{\circ}}{2\left(\cos 40^{\circ}+i \sin 40^{\circ}\right)}$$

Find the vector \(\mathbf{v}\) with the given magnitude and the same direction as \(\mathbf{u}\). $$\begin{array}{cc}\text{Magnitude} && \text{Direction} \\ \|\mathbf{v}\|=8 && \mathbf{u}=\langle 3,3\rangle \end{array}$$

Find \(u \cdot v,\) where \(\theta\) is the angle between u and v. $$\|\mathbf{u}\|=4,\|\mathbf{v}\|=12, \theta=\frac{\pi}{3}$$

Multiplying or Dividing Complex Numbers Perform the operation and leave the result in trigonometric form. $$\frac{\cos \pi+i \sin \pi}{\cos (\pi / 3)+i \sin (\pi / 3)}$$

Determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal. $$\begin{aligned}&\mathbf{u}=\langle-12,30\rangle\\\&\mathbf{v}=\left\langle\frac{1}{2},-\frac{5}{4}\right\rangle\end{aligned}$$

On a baseball diamond with 90-foot sides, the pitcher's mound is 60.5 feet from home plate. How far is it from the pitcher's mound to third base?